inv lve a journal of mathematics Maximal subgroups of the semigroup of partial symmetries of a regular polygon Thomas L. Shelly and Janet E. Mills mathematical sciences publishers 2008 Vol. 1, No. 1 INVOLVE 1:1(2008) Maximal subgroups of the semigroup of partial symmetries of a regular polygon Thomas L. Shelly and Janet E. Mills (Communicated by Scott Chapman) The semigroup of partial symmetries of a polygon P is the collection of all distance-preserving bijections between subpolygons of P, with composition as the operation. Around every idempotent of the semigroup there is a maximal subgroup that is the group of symmetries of a subpolygon of P. In this paper we construct all of the maximal subgroups that can occur for any regular polygon P, and determine for which P there exist nontrivial cyclic maximal subgroups, and for which there are only dihedral maximal subgroups. 1. Introduction and basic properties The semigroup of partial symmetries of a polygon is a natural generalization of the group of symmetries of a polygon. In this paper we will assume knowledge of an undergraduate abstract algebra course and will generally use the terminology of Gallian [2002]. The group of symmetries of a polygon P is the set of distance- preserving mappings of P onto P, with composition as the operation. In particular, for any n > 2, the group of symmetries of a regular n-gon is a group, called the dihedral group, with 2n elements, and is denoted by Dn. The elements of this group are completely determined by the movement of the vertices, and Dn can be considered as a subgroup of the group of all permutations of the vertices of the polygon, under composition. To generalize the notion of symmetries of a polygon, we first need to describe what a subpolygon should be. Let P be a convex polygon with set of vertices V (P) = {v1, v2, . , vn}, listed clockwise, where there exists an edge between vi and vi+1 for i = 1, 2,..., n − 1, and an edge between vn and v1. Let = A vi1 , vi2 , . , vim be a subset of V (P), where i1 < i2 < . < im and let PA be the polygon with edges = − between vi j and vi j+1 for j 1, 2,..., m 1, and between vim and vi1 . The polygon MSC2000: 20M18. Keywords: semigroup, polygon, symmetries. 33 34 THOMAS L. SHELLY AND JANET E. MILLS PA is called a subpolygon of P, and in particular, PA is said to be the subpolygon formed by A. Note that if vi j and vi j+1 are adjacent in P then the edge between them is still an edge in PA; otherwise, the edge between them is new. Note also that the subpolygons include those with no vertices (the empty polygon), exactly one vertex (a point), or two vertices (a line segment). For each subset of V (P), there is a unique subpolygon described since the in- dices on the vertices must be increasing and each subpolygon is a convex polygon. The set of all subpolygons of P will be denoted by 5. Now we must describe the semigroup of partial symmetries of a convex polygon P. This class of semigroups was first defined in [Mills 1990b], and some of its properties explored in [Mills 1990a; 1993]. The domain and range of a function α will be denoted by dom α and ran α respectively. Definition. Let P be a convex polygon. On the set S = S(P) = {α : A → B | PA, PB ∈ 5, and α is a distance-preserving bijection} , define composition by xαβ = (xα)β for all x ∈ domα such that xα ∈ domβ. Then under this operation, S is a semigroup, called the semigroup of partial symmetries of the polygon P. Note that if α is in S and maps A onto B, then α−1 is also a distance-preserving bijection of B onto A, so α−1 is in S. The semigroup S is an example of an inverse semigroup. That is, for each α ∈ S, there is a unique β ∈ S such that αβα = α and βαβ = β. In our semigroup, for α ∈ S, the mapping α−1 serves as the needed β. An idempotent of S is any α such that α2 = α. It is easy to see that because the mappings are one-to-one, α is an idempotent if and only if α is the identity on its domain A, denoted by ιA. In any inverse semigroup the idempotents form a skeleton of the semigroup, and around every idempotent there is a maximal sub- group with that idempotent as its identity. In S, if A is a subset of V (P), then the maximal subgroup with ιA as its identity is HA = {α ∈ S | there exists a β ∈ S such that αβ = βα = ιA} = {α ∈ S | dom α = ran α = A} . This is the largest subgroup of S having ιA as its identity. It is the purpose of this paper to determine, for a regular polygon P, exactly which groups can occur as a maximal subgroup of S(P). It is clear from the description above that the maximal subgroup around ιA is the group of symmetries of the subpolygon PA. Therefore, the effort to find all maximal subgroups of S reduces to describing the group of symmetries of each subpolygon of P. As is well known, the group of symmetries of any polygon is either a dihedral group or a cyclic group [Gallian 2002, Theorem 27.1]. The problem here is that we have MAXIMAL SUBGROUPS OF PARTIAL SYMMETRIES OF A POLYGON 35 a particular regular polygon P, and need to determine exactly which dihedral and cyclic groups can occur as symmetry groups of subpolygons of that polygon P. From now on, we will assume that P is a regular polygon with n vertices. The semigroup S(P) has an identity ιP , and the maximal subgroup around ιP is just the group of symmetries of P, or Dn. As we shall see, this subgroup plays an important part in determining the other maximal subgroups. Therefore, we need to recall some information about the group Dn. In particular, Dn is a group with 2n elements, having n reflections and n rotations. All reflections are about some line of symmetry of P that passes through the center of P. If n is even, every line of symmetry passes through two vertices or through the midpoint between two vertices, whereas if n is odd, every line of symmetry passes through exactly one vertex. Since P is regular, the rotations form a subgroup generated by ρ, which is a rotation about the center of P through 2π/n radians. This subgroup is often written as hρi, the cyclic subgroup generated by ρ, which is isomorphic to ޚn, the group of integers modulo n. In this paper, ρ will always denote this rotation. 2. Maximal subgroups In this section we find all maximal subgroups of S for any regular polygon P. In addition, we provide a description of all subpolygons with rotational symmetry and we give a method for constructing subpolygons with cyclic symmetry groups. For the remainder of the paper we use the following notation: Greek letters are used to represent elements of S, and the letters v and w are used to represent vertices. Thus any expression of the form αβ denotes composition, whereas the expression vα = w says that the vertex v is mapped to w under α (we always write the argument of the function to the left of the function, as in the definition of S in Section 1). The letter d is always used to represent an arbitrary element of Dn. It was shown in [Mills 1993] that for a regular polygon P, every element α ∈ S can be extended to an element in the group of symmetries of P. That is, if dom α = A then α = ιAd for some d ∈ Dn. Further, it was shown that if A has at least 3 elements, then d is unique. For the remainder of the paper, we always take any subset A of V (P) to have more than two elements to ensure every element in the maximal subgroup HA extends uniquely to Dn. Not much is lost by this restriction, since if |A| ≤ 2 then PA is either a point or a line segment, and HA is either ޚ1 or ޚ2. This unique extension guarantees that rotations in HA are extended to rotations in Dn and reflections in HA are extended to reflections in Dn. More specifically, we can connect elements in HA to those in Dn as follows. Lemma 2.1. Let α and β be elements of a maximal subgroup HA, with |A| > 2, such that α = ιAd1 and β = ιAd2 for d1, d2 ∈ Dn. Then the following are true: (a) αβ = ιAd1d2. 36 THOMAS L. SHELLY AND JANET E. MILLS j = j ∈ (b) α ιAd1 for all j ޚ. (c) |α| = |d1|, where |α| and |d1| are the orders of α and d1 in HA and Dn respec- tively. Proof. To prove Lemma 2.1a, let α and β be defined as above. Then since ιA is the identity in HA, αβ = (ιAd1)(ιAd2) = ((ιAd1) ιA) d2 = (ιAd1) d2 = ιAd1d2. The proof of Lemma 2.1b is a simple application of Lemma 2.1a using induction and −1 = −1 | | = the fact that α ιAd1 . To prove Lemma 2.1c, suppose that α m. Then m is m = m = m the smallest positive integer such that α ιA. From Lemma 2.1b, α ιAd1 , so = m | | ιA ιAd1 .